How would I animate vertices randomly up and down (+looping)
There are multiple ways to do it, the simplest I can come up is to set an empty as object in the displace modifier, animate the empty and make the first and last frame the same, then make "cyclic" extrapolation of the animation 
If you need add randomness, you can consider the noise f-curve modifier 

First example with 3 keyframes and the noise modifier



The second example use Make Cyclic f-modifier and only two keyframe



Not sure why in the second example the loop works only twice if someone knows please advice, either it's a bug or it's a matter of mapping (I'm to tired now to debug)